Water Injector Well – Importance of Inertial Effects

This two-dimensional model depicts a vertical well coupled to a sandstone aquifer. The well is cased and perforated, and water represents the single phase. The well is completed with a concentric screen, modeled here as a porous medium. The annular gap is filled with a sand pack (shown in green in Figure 1) up to three quarters of the considered well length. Water is injected at constant pressure through the top of the base pipe, which feeds into the screen pipe. The equidistant perforation tunnels are also filled with a sand pack having the same properties as that in the annulus.

The equations solved for are the stationary Navier-Stokes in the free domains (depicted in blue in the same Figure), and the Brinkman equations for the porous ones (screen, sand pack, and aquifer matrix). The aquifer is arbitrarily divided into the near and far domains. Inertial effects are considered for the sand pack, for which both beta factor and permeability are described via the respective Ergun correlations. All materials are considered incompressible and at constant temperature.

The applied boundary conditions are constant injection pressure (Pinlet) at the base pipe inlet, constant zero pressure (Poutlet) at the aquifer far field outlet, and no flow on the other outer boundaries of the model. The wellbore connects hydraulically to the aquifer only through the sand-packed perforation tunnels.

Figure 1. Meshed domains. The Navier-Stokes equations are solved in the free domains (depicted in blue), while the flow in porous media is described by the Brinkman equation. Inertial effects are also considered for the sand-packed annulus and perforation tunnels (shown in green). The inset (B) provides a close-up of the top-left section of the completion, indicated by the dashed magenta rectangle.

The length of the well is 1.5 m, the outer and inner diameters of the screen and casing are 3.32 inch and 6.63 inch, respectively, and the modeled aquifer extends to 15 ft away from the axis of the well. The borehole has a diameter of 9.5 inch, while the seven perforation tunnels are 0.75 inch-wide. The screen assembly hydraulically connects the interior of the screen pipe to the well annulus.

The sand pack consists of particles 400 micron in diameter, with a porosity of 30%. The Ergun correlations lead therefore to a permeability Ksand = 59.55 Darcy and a beta factor of 1.13E5 m-1. The permeability and porosity of the formation are 100 mD and 20%, respectively. The screen assembly is modeled as a porous medium of porosity equal to unity and 1000 Darcy permeability, without inertial effects.

The results below were obtained with a parametric stationary study for given injection pressure values. The quantities reported are the pressure differentials and superficial velocities across the top, middle and bottom perforations, and fluid pressure values (relative to that of the undisturbed formation, set as constant to zero at the far field boundary) across selected vertical boundaries.

Figure 2 compares the domain-averaged superficial velocity values encountered through the the top (blue), middle (green), and bottom (red) perforation tunnels, with inertial effects included. The superficial velocity represents a proxy for the perforation tunnel flow rate. The top perforation tunnel takes in the most fluid, thanks to its proximity to the top of the sand pack. Perhaps somewhat counter-intuitively, the middle tunnel takes in the least, as it competes with the nearby perforation tunnels.

Figure 2. Superficial velocity values across the top (blue), middle (green), and bottom (red) perforation tunnels, with inertial effects considered for the sand pack.

Figures 3 and 4 show pressure differential values across the top (blue), middle (green), and bottom (red) perforation tunnels, with and without inertial effects, respectively. Correlating with the results shown above, the pressure drop across the top perforation is the largest. Apparent is the effect (in Figure 3) of including the inertial term in the flow model: the pressure differential values becomes increasingly non-linear for higher superficial velocities. Figure 5 exemplifies this departure for the top perforation.

Figure 3. Pressure differential values across the top (blue), middle (green), and bottom (red) perforation tunnels, with inertial effects considered for the sand pack.

Figure 4. Pressure differential values across the top (blue), middle (green), and bottom (red) perforation tunnels, in the absence of inertial effects. Notice the linear behavior throughout the injection pressure range.

Figure 5. Pressure differential values across the top perforation tunnel, with and without inertial effects (solid blue and dashed cyan curves, respectively). Notice the non-linear behavior evident at the higher values of the injection pressure.

Another interesting aspect is how the fluid pressure varies vertically at different radial distances from the well axis. Figure 6 displays (for Pinlet =1000 psi) pressure values along selected vertical edges: on the inner and outer walls of the cement-casing assembly (blue and green curves, respectively), and 1.5 ft away from the well axis (red curve, representing the near-far aquifer division line). The locations of the peaks on the green curve correspond to those of the perforation tunnels, with the left-most peak centered on the top perforation.

Figure 6. Pressure values relative to that of the undisturbed aquifer, along selected vertical edges: the casing inner wall (blue), the borehole wall (green), and 1.5 ft away from the well axis (within the aquifer). The peaks apparent on the green curve center onto the perforation tunnel locations, with the left-most peak corresponds to the top perforation.